Abstract

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh–Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.

Lead Paragraph: The spontaneous onset of synchronization in coupled nonlinear oscillators is a remarkable and ubiquitous phenomenon. In recent years there has been much effort to explore various types of synchronization, including phase synchronization, when the dynamics is correlated in phase but uncorrelated in amplitude. Phase synchronization has been increasingly studied in the context of potential applications in disciplines ranging from physics and chemistry to biology and medical sciences. It was found that the degree of synchronization can be an important part of the function or malfunction of a given system, and in many cases synchronization in phase turned out to be undesirable. In mechanical systems synchronization may result in dangerous jams or overloads. Similarly, several neurological disorders such as epileptic seizures or Parkinson’s disease are associated with synchronized firings of neurons, while in ecological systems, the synchronization of populations is often seen as detrimental because it enhances the chances of global species extinctions. These findings highlight the need to explore reliable methods for preventing the formation of phase synchronization in coupled oscillatory systems. One such possibility was recently identified if the coupling between the subsystems allows for time delay. In this case the system can undergo a phase-flip bifurcation,1,2 where the coupled oscillators alternate from a state of in-phase to antiphase, with the emergence of large phase differences between the interacting systems. Here we demonstrate that the phase-flip bifurcation occurs in a wide and important class of systems, including excitable dynamics, that apply in laser and neuronal systems, and regular and chaotic cycling ecological models. Further we provide the first explicit experimental verification of this bifurcation in coupled electronic circuits. Taken together, our results suggest the phase-flip bifurcation to be a general and important property of time-delay coupled nonlinear systems.

Acknowledgments:

A.P. and S.K.D. are supported by the DST, India. R.K. is the recipient of a Junior Research Fellowship from the UGC, India. A.P. would like to thank the University of Potsdam for support and hospitality. J.K. acknowledges support from the Humboldt Foundation, Germany, and the CSIR, India.